Calculus Examples

$y = 3 t^{7} - 6 t^{4} + 8 t^{3} - 12^{t} + 18$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (y) = \frac{d}{d t} (3 t^{7} - 6 t^{4} + 8 t^{3} - 12^{t} + 18)$

Step 2

The derivative of $y$ with respect to $t$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $3 t^{7} - 6 t^{4} + 8 t^{3} - 12^{t} + 18$ with respect to $t$ is $\frac{d}{d t} [3 t^{7}] + \frac{d}{d t} [- 6 t^{4}] + \frac{d}{d t} [8 t^{3}] + \frac{d}{d t} [- 12^{t}] + \frac{d}{d t} [18]$ .

$\frac{d}{d t} [3 t^{7}] + \frac{d}{d t} [- 6 t^{4}] + \frac{d}{d t} [8 t^{3}] + \frac{d}{d t} [- 12^{t}] + \frac{d}{d t} [18]$

Step 3.2

Evaluate $\frac{d}{d t} [3 t^{7}]$ .

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Step 3.2.1

Since $3$ is constant with respect to $t$ , the derivative of $3 t^{7}$ with respect to $t$ is $3 \frac{d}{d t} [t^{7}]$ .

$3 \frac{d}{d t} [t^{7}] + \frac{d}{d t} [- 6 t^{4}] + \frac{d}{d t} [8 t^{3}] + \frac{d}{d t} [- 12^{t}] + \frac{d}{d t} [18]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 7$ .

$3 (7 t^{6}) + \frac{d}{d t} [- 6 t^{4}] + \frac{d}{d t} [8 t^{3}] + \frac{d}{d t} [- 12^{t}] + \frac{d}{d t} [18]$

Step 3.2.3

Multiply $7$ by $3$ .

$21 t^{6} + \frac{d}{d t} [- 6 t^{4}] + \frac{d}{d t} [8 t^{3}] + \frac{d}{d t} [- 12^{t}] + \frac{d}{d t} [18]$

Step 3.3

Evaluate $\frac{d}{d t} [- 6 t^{4}]$ .

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Step 3.3.1

Since $- 6$ is constant with respect to $t$ , the derivative of $- 6 t^{4}$ with respect to $t$ is $- 6 \frac{d}{d t} [t^{4}]$ .

$21 t^{6} - 6 \frac{d}{d t} [t^{4}] + \frac{d}{d t} [8 t^{3}] + \frac{d}{d t} [- 12^{t}] + \frac{d}{d t} [18]$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 4$ .

$21 t^{6} - 6 (4 t^{3}) + \frac{d}{d t} [8 t^{3}] + \frac{d}{d t} [- 12^{t}] + \frac{d}{d t} [18]$

Step 3.3.3

Multiply $4$ by $- 6$ .

$21 t^{6} - 24 t^{3} + \frac{d}{d t} [8 t^{3}] + \frac{d}{d t} [- 12^{t}] + \frac{d}{d t} [18]$

Step 3.4

Evaluate $\frac{d}{d t} [8 t^{3}]$ .

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Step 3.4.1

Since $8$ is constant with respect to $t$ , the derivative of $8 t^{3}$ with respect to $t$ is $8 \frac{d}{d t} [t^{3}]$ .

$21 t^{6} - 24 t^{3} + 8 \frac{d}{d t} [t^{3}] + \frac{d}{d t} [- 12^{t}] + \frac{d}{d t} [18]$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 3$ .

$21 t^{6} - 24 t^{3} + 8 (3 t^{2}) + \frac{d}{d t} [- 12^{t}] + \frac{d}{d t} [18]$

Step 3.4.3

Multiply $3$ by $8$ .

$21 t^{6} - 24 t^{3} + 24 t^{2} + \frac{d}{d t} [- 12^{t}] + \frac{d}{d t} [18]$

Step 3.5

Evaluate $\frac{d}{d t} [- 12^{t}]$ .

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Step 3.5.1

Since $- 1$ is constant with respect to $t$ , the derivative of $- 12^{t}$ with respect to $t$ is $- \frac{d}{d t} [12^{t}]$ .

$21 t^{6} - 24 t^{3} + 24 t^{2} - \frac{d}{d t} [12^{t}] + \frac{d}{d t} [18]$

Step 3.5.2

Differentiate using the Exponential Rule which states that $\frac{d}{d t} [a^{t}]$ is $a^{t} \ln (a)$ where $a$ = $12$ .

$21 t^{6} - 24 t^{3} + 24 t^{2} - 12^{t} \ln (12) + \frac{d}{d t} [18]$

Step 3.6

Differentiate using the Constant Rule.

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Step 3.6.1

Since $18$ is constant with respect to $t$ , the derivative of $18$ with respect to $t$ is $0$ .

$21 t^{6} - 24 t^{3} + 24 t^{2} - 12^{t} \ln (12) + 0$

Step 3.6.2

Add $21 t^{6} - 24 t^{3} + 24 t^{2} - 12^{t} \ln (12)$ and $0$ .

$21 t^{6} - 24 t^{3} + 24 t^{2} - 12^{t} \ln (12)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 21 t^{6} - 24 t^{3} + 24 t^{2} - 12^{t} \ln (12)$

Step 5

Replace $y'$ with $\frac{d y}{d t}$ .

$\frac{d y}{d t} = 21 t^{6} - 24 t^{3} + 24 t^{2} - 12^{t} \ln (12)$